9780387989532

Probability for Statisticians

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  • ISBN13:

    9780387989532

  • ISBN10:

    0387989536

  • Format: Hardcover
  • Copyright: 2000-07-01
  • Publisher: Springer Nature
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Summary

Probability for Statisticians is intended as a text for a one year graduate course aimed especially at students in statistics. The choice of examples illustrates this intention clearly. The material to be presented in the classroom constitutes a bit more than half the text, and the choices the author makes at the University of Washington in Seattle are spelled out. The rest of the text provides background, offers different routes that could be pursued in the classroom, ad offers additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Stein's method either prior to or alternative to a characteristic funcion presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function. The bootstrap and trimming are both presented. The martingale coverage includes coverage of censored data martingales. The text includes measure theoretic preliminaries, from which the authors own course typically includes selected coverage. The author is a professor of Statistics and adjunct professor of Mathematics at the University of Washington in Seattle. He served as chair of the Department of Statistics 1986-- 1989. He received his PhD in Statistics from Stanford University. He is a fellow of the Institute of Mathematical Statistics, and is a former associate editor of the Annals of Statistics.

Table of Contents

Preface vii
Use of This Text xiii
Definition of Symbols xviii
Measures
Basic Properties of Measures
1(11)
Construction and Extension of Measures
12(6)
Lebesgue-Stieltjes Measures
18(3)
Measurable Functions and Convergence
Mappings and σ-Fields
21(3)
Measurable Functions
24(5)
Convergence
29(4)
Probability, RVs, and Convergence in Law
33(2)
Discussion of Sub σ-Fields
35(2)
Integration
The Lebesgue Integral
37(3)
Fundamental Properties of Integrals
40(4)
Evaluating and Differentiating Integrals
44(2)
Inequalities
46(5)
Modes of Convergence
51(10)
Derivatives via Signed Measures
Decomposition of Signed Measures
61(5)
The Radon-Nikodym Theorem
66(4)
Lebesgue's Theorem
70(4)
The Fundamental Theorem of Calculus
74(5)
Measures and Processes on Products
Finite-Dimensional Product Spaces
79(5)
Random Vectors on (Ω, A, P)
84(2)
Countably Infinite Product Probability Spaces
86(4)
Random Elements and Processes on (Ω, A, P)
90(5)
General Topology and Hilbert Space
General Topology
95(6)
Metric Spaces
101(3)
Hilbert Space
104(3)
Distribution and Quantile Functions
Character of Distribution Functions
107(3)
Properties of Distribution Functions
110(1)
The Quantile Transformation
111(4)
Integration by Parts Applied to Moments
115(4)
Important Statistical Quantities
119(4)
Infinite Variance
123(5)
Slowly Varying Partial Variance
128(7)
Specific Tail Relationships
135(3)
Regularly Varying Functions
138(4)
Some Winsorized Variance Comparisons
142(7)
Inequalities for Winsorized Quantile Functions
149(4)
Independence and Conditional Distributions
Independence
153(4)
The Tail σ-Field
157(2)
Uncorrelated Random Variables
159(1)
Basic Properties of Conditional Expectation
160(10)
Regular Conditional Probability
170(6)
Conditional Expectations as Projections
176(5)
Special Distributions
Elementary Probability
181(8)
Distribution Theory for Statistics
189(4)
Linear Algebra Applications
193(8)
The Multivariate Normal Distribution
201(4)
WLLN, SLLN, LIL, and Series
Introduction
205(1)
Borel-Cantelli and Kronecker Lemmas
206(2)
Truncation, WLLN, and Review of Inequalities
208(4)
Maximal Inequalities and Symmetrization
212(5)
The Classical Laws of Large Numbers, LLNs
217(8)
Applications of the Laws of Large Numbers
225(3)
General Moment Estimation
228(9)
Law of the Iterated Logarithm
237(4)
Strong Markov Property for Sums of IID RVs
241(2)
Convergence of Series of Independent RVs
243(5)
Martingales
248(1)
Maximal Inequalities, Some with → Boundaries
249(5)
A Uniform SLLN
254(3)
Convergence in Distribution
Stein's Method for CLTs
257(9)
Winsorization and Truncation
266(5)
Identically Distributed RVs
271(5)
Bootstrapping
276(2)
Bootstrapping with Slowly Increasing Trimming
278(3)
Examples of Limiting Distributions
281(9)
Classical Convergence in Distribution
290(4)
Limit Determining Classes of Functions
294(3)
Brownian Motion and Empirical Processes
Special Spaces
297(3)
Existence of Processes on (C,C) and (D,D)
300(4)
Brownian Motion and Brownian Bridge
304(3)
Stopping Times
307(3)
Strong Markov Property
310(3)
Embedding a RV in Brownian Motion
313(3)
Barrier Crossing Probabilities
316(4)
Embedding the Partial Sum Process
320(5)
Other Properties of Brownian Motion
325(2)
Various Empirical Processes
327(8)
Inequalities for the Various Empirical Processes
335(5)
Applications
340(3)
Characteristics Functions
Basic Results, with Derivation of Common Chfs
343(5)
Uniqueness and Inversion
348(4)
The Continuity Theorem
352(2)
Elementary Complex and Fourier Analysis
354(6)
Esseen's Lemma
360(3)
Distributions of Grids
363(2)
Conditions for ø to Be a Characteristic Function
365(2)
CLTs via Characteristic Functions
Introduction
367(1)
Basic Limit Theorems
368(5)
Variations on the Classical CLT
373(9)
Local Limit Theorems
382(3)
Gamma Approximation
385(8)
Edgeworth Expansions
393(6)
Approximating the Distribution of h(Xn)
399(2)
Infinitely Divisble and Stable Distributions
Infinitely Divisible Distributions
401(8)
Stable Distributions
409(3)
Characterizing Stable Laws
412(2)
The Domain of Attraction of a Stable Law
414(3)
Asymptotics via Empirical Proceses
Introduction
417(1)
Trimmed and Winsorized Means
418(10)
Linear Rank Statistics and Finite Sampling
428(6)
The Bootstrap
434(5)
L-Statistics
439(12)
Asymptotics via Stein's Approach
U-Statistics
451(9)
Hoeffding's Combinatorial CLT
460(9)
Martingales
Basic Technicalities for Martingales
469(5)
Simple Optional Sampling Theorem
474(1)
The Submartingale Convergence Theorem
475(8)
Applications of the S-mg Convergence Theorem
483(6)
Decomposition of a Submartingale Sequence
489(5)
Optional Sampling
494(7)
Applications of Optional Sampling
501(2)
Introduction to Counting Process Martingales
503(10)
Doob-Meyer Submartingale Decomposition
513(5)
Predictable Processes and ∫0 H dM Martingales
518(6)
The Basic Censored Data Martingale
524(7)
CLTs for Dependent RVs
531(2)
Convergence in Law on Metric Spaces
Convergence in Distribution on Metric Spaces
533(9)
Metrics for Convergence in Distribution
542(6)
Appendix A. Distribution Summaries
1. The Gamma and Digamma Functions
548(5)
2. Maximum Likelihood Estimators and Moments
553(4)
3. Examples of Statistical Models
557(8)
4. Asymptotics of Maximum Likelihood Estimation
565(5)
References 570(5)
Index 575

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